Design of Engineering Experiments Hussam Alshraideh Chapter 2: Some Basic Statistical Concepts October 4, 2015 Hussam Alshraideh (JUST) Basic Stats October 4, 2015 1 / 29
Overview 1 Introduction Basic probability concepts Common probability distributions 2 Statistical Inference Estimation Hypothesis testing Normal Probability Plots Hussam Alshraideh (JUST) Basic Stats October 4, 2015 2 / 29
Introduction Basic probability concepts Basic concepts Random experiment: an experiment whose outcome is not known in advance. Flipping a coin {H, T} Throwing a dice {1,2,3,4,5,6} Two dice {(1,1),(1,2),,(6,6)} Student height Sample space (S): the set of all possible outcomes of a random experiment. Flipping a coin S = {H, T } Throwing a dice S = {1, 2, 3, 4, 5, 6} Two dice S = {(1, 1), (1, 2),, (6, 6)} Student height S = {100 h 250} Continuous vs. Discrete sample space. Hussam Alshraideh (JUST) Basic Stats October 4, 2015 3 / 29
Introduction Basic probability concepts Basic concepts Event: a subset of the sample space. Coin: E = {H} Dice: E = {1, 2} 2 dice: sum 3 E = {(1, 1), (1, 2), (2, 1)} Probability of an event P(E): the likelihood of observing the event E. P(E) = # of elements in E # of elements in S Coin: E={H}, P(E) = 1 2 = 0.5 Dice: E={1,2}, P(E) = 2 6 2 dice: sum 3 E={(1,1),(1,2),(2,1)}, P(E) = 3 36 Hussam Alshraideh (JUST) Basic Stats October 4, 2015 4 / 29
Introduction Basic probability concepts Basic concepts Random variable: a real valued function defined over the sample space of a random experiment. Coin: { 1, H P(X = 1) = P({H}) = 0.5 X = 0, T P(X = 0) = P({T }) = 0.5 2 dice: X = sum of two numbers, X {2, 3,, 12} P(X = 3) = P({(1, 2), (2, 1)}) = 2 36 Student height: X = height, 100 X 250 Continuous vs. Discrete random variables. Hussam Alshraideh (JUST) Basic Stats October 4, 2015 5 / 29
Introduction Basic probability concepts Basic concepts Discrete random variables are described by their Probability Mass Function (pmf). X 1 5 6 10 P(X) 0.1 0.3 0.1 0.5 Hussam Alshraideh (JUST) Basic Stats October 4, 2015 6 / 29
Introduction Basic probability concepts Basic concepts Probability density function (pdf): a function that is used to determine probabilities of continuous random variable from the area under the function. P(a X b) = b a f (x)dx such that: f (x) 0, f (x)dx = 1 Hussam Alshraideh (JUST) Basic Stats October 4, 2015 7 / 29
Introduction Basic probability concepts Basic concepts Expected value E(X ): Discrete r.v. µ = E(X ) = x xp(x) Continuous r.v. µ = E(X ) = xf (x)dx Hussam Alshraideh (JUST) Basic Stats October 4, 2015 8 / 29
Introduction Basic probability concepts Basic concepts Variance Var(X ): Discrete r.v. σ 2 = Var(X ) = x (x µ) 2 p(x) Continuous r.v. σ 2 = Var(X ) = (x µ) 2 f (x)dx Var(X ) = σ 2 = E(X 2 ) (E(X )) 2 = E(X 2 ) µ 2 Hussam Alshraideh (JUST) Basic Stats October 4, 2015 9 / 29
Introduction Common probability distributions The Normal distribution f (x) = 1 2πσ e (x µ)2 2σ 2, x Mean µ and variance σ 2 are called distribution parameters. µ controls the location, while σ 2 controls the shape. Hussam Alshraideh (JUST) Basic Stats October 4, 2015 10 / 29
Introduction Common probability distributions The Normal distribution Special case when µ = 0 and σ = 1, the standard normal distribution. f (x) = 1 2π e x2 2, x Probabilities under the standard normal curve are given in tables at the end of the textbook. Any normal r.v. can be transformed to a standard normal r.v. using: z = x µ σ Hussam Alshraideh (JUST) Basic Stats October 4, 2015 11 / 29
Introduction Common probability distributions The Normal distribution Example: The diameter of a shaft in a storage drive is normally distributed with mean 0.2508 inch and standard deviation 0.0005 inch. The specifications on the shaft are 0.2500±0.0015 inch. What proportion of shafts conforms to specifications? Hussam Alshraideh (JUST) Basic Stats October 4, 2015 12 / 29
Estimation Population vs. sample The totality of all observations of a random variable is the population. A portion used for analysis is a random sample. A population is described, in part, by its parameters, i.e., mean (µ) and standard deviation (σ). A random sample of size n is drawn from a population and is described, in part, by its statistics, i.e., mean (x-bar) and standard deviation (s). The statistics are used to estimate the parameters. Hussam Alshraideh (JUST) Basic Stats October 4, 2015 13 / 29
Estimation Point estimation A point estimate of some population parameter θ is a single numerical value θ of a statistic Θ. Hussam Alshraideh (JUST) Basic Stats October 4, 2015 14 / 29
Estimation Point estimation The point estimator Θ is an unbiased estimator for the parameter θ if: E( Θ) = θ if not unbiased, then the quantity : E( Θ) θ is called the bias. Sampling distribution: the distribution of a statistic. Hussam Alshraideh (JUST) Basic Stats October 4, 2015 15 / 29
Hypothesis testing Hypothesis testing A statistical hypothesis is a statement about the parameters of one or more populations. H 0 : µ = 50 H 1 : µ 50 Type I error: rejecting the null hypothesis H 0 when it is true. Type II error: failing to reject the null hypothesis when it is false. α = p(type I error), β = p(type II error) Hussam Alshraideh (JUST) Basic Stats October 4, 2015 16 / 29
Hypothesis testing Hypothesis testing Note: see handout of hypothesis testing procedures. Hussam Alshraideh (JUST) Basic Stats October 4, 2015 17 / 29
Hypothesis testing Inference on the mean of a population, variance known Given X N(µ, σ 2 ) where µ in unknown and σ 2 is known. Test the hypothesis: H 0 : µ = µ 0 H 1 : µ µ 0 under normality and independent samples assumptions, the statistic z = x µ 0 σ/ n N(0, 1) Idea: calculate z for the given sample, then find the probability that it came from the N(0, 1) distribution. If this probability is large, then accept H 0. If not, then reject H 0. Large enough is determined by 1 α. Hussam Alshraideh (JUST) Basic Stats October 4, 2015 18 / 29
Hypothesis testing Inference on the mean of a population, variance known Example: Aircrew escape systems are powered by a solid propellant. The burning rate of this propellant is an important product characteristic. Specifications require that the mean burning rate must be 50 cm/s. We know that the standard deviation of burning rate is σ = 2 cm/s. The experimenter decides to specify a type I error probability or significance level of α = 0.05. He selects a random sample of n = 25 and obtains a sample average burning rate of x = 51.3 cm/s. What conclusions should he draw? Solution: Test z 0 = H 0 : µ = 50 H 1 : µ 50 51.3 50 2/ 25 = 3.25 SInce z 0 > z α/2 = z 0.025 = 1.96, then reject H 0. Hussam Alshraideh (JUST) Basic Stats October 4, 2015 19 / 29
Hypothesis testing Inference on the mean of a population, variance known Can get the same result using the P-value. P-value=Probability that the distribution under the null hypothesis produces a value that is as extreme as the test statistic. P-value=p(z z0) Hussam Alshraideh (JUST) Basic Stats October 4, 2015 20 / 29
Hypothesis testing Two sample t-test Example: An engineer is studying the formulation of a Portland cement mortar. He has added a polymer latex emulsion during mixing to determine if this impacts the curing time and tension bond strength of the mortar. The experimenter prepared 10 samples of the original formulation and 10 samples of the modified formulation. Hussam Alshraideh (JUST) Basic Stats October 4, 2015 21 / 29
Hypothesis testing Two sample t-test Hussam Alshraideh (JUST) Basic Stats October 4, 2015 22 / 29
Hypothesis testing Two sample t-test Test: H 0 : µ 1 = µ 2 H 1 : µ 1 µ 2 Hussam Alshraideh (JUST) Basic Stats October 4, 2015 23 / 29
Hypothesis testing Two sample t-test Test statistic: Statistic Modified mortar Unmodified mortar y 16.76 17.04 S 2 0.1 0.061 S 0.316 0.248 n 10 10 t 0 = y 1 y 2, S S 1 p n 1 + 1 p 2 = (n 1 1)S 2 1 + (n 2 1)S2 2 n 1 + n 2 2 n 2 Note: this is a Signal to Noise ratio. 16.76 17.04 t 0 = 0.284 Hence, reject H 0. 1 10 + 1 10 = 2.20, t α/2,n1 +n 2 2 = t 0.025,18 = 2.101 Hussam Alshraideh (JUST) Basic Stats October 4, 2015 24 / 29
Normal Probability Plots Assumptions 1 Data is normally distributed, Check using Normal Probability Plots (NPP) 2 Independent samples. No need to validate if random samples are used. Hussam Alshraideh (JUST) Basic Stats October 4, 2015 25 / 29
Normal Probability Plots Normal Probability Plot To construct a Normal Probability Plot (NPP): 1 Sort the data observations in an ascending order: x (1), x (2),, x (n). 2 The observed value x (j) is plotted against the cumulative distribution (j 0.5). n 3 If the paired numbers form a straight line, it is reasonable to assume that the data follows the proposed distribution. Hussam Alshraideh (JUST) Basic Stats October 4, 2015 26 / 29
Normal Probability Plots Normal Probability Plot: Example Hussam Alshraideh (JUST) Basic Stats October 4, 2015 27 / 29
Normal Probability Plots Homework Solve (both manually and using Minitab) the following problems from the end of chapter 2 problems in the textbook. 1 Problem 2.4 2 Problem 2.9 3 Problem 2.22 4 Problem 2.25 5 Problem 2.32 6 Problem 2.35 Hussam Alshraideh (JUST) Basic Stats October 4, 2015 28 / 29
Normal Probability Plots Question What if more than two samples? Hussam Alshraideh (JUST) Basic Stats October 4, 2015 29 / 29